Scott Sinclair opened the scoring after just six minutes before Ashley Williams doubled the advantage and Moore netted his first goal for the club in the 78th minute.
Rodgers said: "A goalscorer not scoring is like a hungry person without food. It's been difficult for Luke but he has kept working hard and has the trust of his team-mates.
"But the goal should relieve the pressure on him. Hopefully it will give his confidence a boost and he can kick on and score many more goals for us.
"The performance was excellent and it shows the progress we have made. I thought Ashley Williams and Alan Tate were outstanding but there is always room for improvement.
"It was good to get an early goal after their result in midweek. Scoring then was probably a setback mentally for them. You can see why Scott Sinclair has scored so many goals because of his desire to get in the box. We have moved up one place in the table which is always good to see."
Source: PA
Source: PA